An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 82.

Branching points.

133

p =- 0. Therefore also its values that belong to points arbitrarily
near the positive axis of abscissa with positive ordinates, differ by
finite quantities from the values possessed by points below that
axis. A branch of the function constructed in this way is therefore discontinuous along the positive axis of abscissa. It is usual
also to state the matter geometrically thus: A branch of the function
q is continuous in the connected surface that consists of the infinite
plane perforated from its zero point to its infinity point.
It is in fact easily seen that everywhere else the branch is not
only unique, as follows from the construction, but also continuous.
For, z = (cos P + - sin!p) being any point for which lP differs finitely
from 0 or 2 z, let us surround it with a small circle of radius A; the
coordinates of points upon or within the circumference of this circle are
z + A+ =- + - (cos 0 +- i sin 0) == ros s +- i sin A9)
(E < A, 0<0 < 2z),
so that:
r cos 9 c Q s s cos, rsin 9 == Q sin p + sin Q.
Hence follows that r = ' q + 2 + - 2E cos (g - 0); therefore AV
can be chosen so as to make the difference abs [r - Q] less than an
arbitrarily small quantity 6, whence it follows further that we can
also put 9p _== + A, where q is arbitrarily small. If now 4p differ
from 0 or 2 z by a finite quantity, ~t + is always a positive number
between zero and 2z, and the values of the function:
wt + Awv = (C 4 (cos P ( +,) + i sin - ( + 4 ))
differ, as the respective series show, arbitrarily little from those of:
=w (cos + i sin ).
This method of rendering the function w unique and continuous,
by drawing a section that must not be crossed by the argument z as
it varies, was introduced by Cauchy. Riemann perfected it by
a process which enables us to contemplate simultaneously all branches,
and to render the function unique and continuous along all paths
without restriction. This is effected, for a function admitting of q
values, by making the variable z move upon q different plane leaves.
We shall first consider the simplest case, assuming q = 2. Besides
the one plane perforated along the positive axis of x, to which we
have coordinated the values of the function starting from the values
w == rT, we take a second plane for the motion of z. In this,
conformably with the general equation: